![𝑅 [Å] 𝑅 [Å]𝑅 [Å]
Calculating transition amplitudes by variational quantum eigensolvers
C71.00241 : Implementation of excited state energy and its analytical derivatives for photochemical reaction simulations on NISQ devices
Yohei Ibe1, Yuya O. Nakagawa1,Takahiro Yamamoto1, Kosuke Mitarai2, 1, Tennin Yan1, Qi Gao3, Takao Kobayashi3
1QunaSys Inc., 2Osaka University, 3Mitsubishi Chemical Corp.
1. Methods for excited states based on VQE
ü Simulating excited states of molecules is of great concern in photophysics & photochemistry,
and there are several methods based on a quantum algorithm, Variational Quantum Eigensolver (VQE)
ü Among them, VQD method is remarkably accurate; however, there are no methods to calculate
transition amplitudes (i.e., off-diagonal matrix elements) between eigenstates obtained by the VQD
ü We propose such a method feasible on NISQ devices and demonstrate on a sampling simulator
Summary
3. Calculating transition amplitudes
ü VQD is an accurate way to simulate excited states on a quantum
computer
ü We proposed a method to calculate transition amplitudes
between two orthogonal states in a hardware-friendly manner,
which is applicable (not only) for the VQD
ü This work enlarges the possibility of the VQD and advances the
field of excited states calculations on a quantum device
4. Sampling simulation with shot noise
5. Conclusion
Appendix: Comparison of ansatz
LiH
𝑅
Li H
STO-3G
(4e, 6o)
(12 qubits)
Method for
excited states
Accuracy
Easy to calculate
transition amplitudes?
SSVQE Fair Yes
MCVQE Intermediate Yes
VQD Excellent No Yes (This work)
Diazene (N2H2)
2. Comparison of methods by noiseless simulation
Potential energy curve Error from exact calc.
O. Higgott et al., Quantum 3, 159 (2019)R. Parrish et al.,
Phys. Rev. Lett. 122, 230401 (2019)
K. Nakanishi et al.,
Phys. Rev. Research 1, 033062 (2019)
ℒ(𝜽) = *
+,-
.
𝑤+⟨𝜑+ 𝑈3
𝜽 𝐻𝑈 𝜽 𝜑+⟩
(𝑤-> 𝑤7 > ⋯ > 0)
ℒ(𝜽) = *
+,-
.
⟨𝜑+ 𝑈3
𝜽 𝐻𝑈 𝜽 𝜑+⟩
1. Obtain ground state E; with VQE
2. Execute VQE routine to minimize
ℒ- 𝜃 = 𝜓 𝜃 𝐻 𝜓 𝜃 + 𝑤- ⟨E;|𝜓 𝜃 ⟩ 7
→ Obtain first excited state E- = 𝜓 𝜃-
∗
3. Repeat for higher excited states
SSVQE
Subspace-Search VQE
MCVQE
Multistate-Contracted VQE
VQD
Variational Quantum Deflation
: physical quantity (Hermitian op.), where .
Following equality holds for transition amplitude :
Each term can be measured on real devices
Calculation setups
• Molecular structures: several points along minimum energy
path between S2 Franck-Condon (cis/trans) & S2 minimum
• Calculation level: 6-31G*/CASCI(3o, 4e), 8 qubits
• Ansatz: RSP ansatz (D=20, see Appendix)
• High-speed simulator Qulacs [http://qulacs.org/] is used
Enables calculation of transition amplitudes with VQD
Unable to measure
as it is (A is not unitary)
Definition of oscillator strength 𝑓+B
: 𝛼-coordinate of the 𝑙-th electron
where
: electric dipole moment operator
in atomic units
LiH
𝑅
Li H
STO-3G
CASCI(2e, 2o)
(2 qubits)
Potential energy curve & error Oscillator strength
RY ansatz
ü VQD can generate excited states the most accurately
ü SSVQE and MCVQE can readily calculate transition amplitudes
ü However, VQD has no known methods to calculate transition
amplitudes on real NISQ devices
ü Transition amplitudes are required for calculating
various physical quantities (e.g., oscillator strengths)
(𝐴: Hermitian op.)
Calculation setups
• Calculation level: 6-31G*/CASCI(2o, 2e), 2 qubits (parity mapping is used to reduce the number of qubits)
• Ansatz: RY ansatz (D=2)
• Two options for the optimization routine for this experiment
Blue dots: use sampling simulator in the whole process (including optimization routine)
Orange dots: use sampling simulator only for calculating oscillator strengths (parameters are optimized
with the noiseless simulator), still using our proposed method
Using RSP ansatz (D=10, see Appendix)
J. T. Seeley et al., J. Chem. Phys. 137, 224109 (2012)
(Available in Qiskit Aqua v0.6.4)
Unitary operators
Assuming
Hardware-efficient ansatz
𝑈FGGHI 𝜽 = exp 𝑇HI(𝜽) − 𝑇HI
3
(𝜽)
𝑇HI 𝜽 = *
+:PQQRSTUV
W:XTYZR[
𝜃+
W
𝑐W
3
𝑐+
+ *
+B:PQQRSTUV
W^:XTYZR[
𝜃+B
W^
𝑐W
3
𝑐^
3
𝑐+ 𝑐B
A. Kandala et al., Nature 549, 242 (2017) A. Peruzzo et al., Nat. Comm., 5,4213 (2014)
×𝐷
UCC-SD ansatz RSP ansatz
(real-valued symmetry preserving ansatz)
𝑅 [Å] 𝑅 [Å]
𝑅 [Å]
𝑅 [Å]
Energy[Ha]
For original symmetry-preserving ansatz, see
P. Barkoutsos et al., Phys. Rev. A 98, 022322 (2018)
Energy[Ha]
Energy[Ha]
Molecule: LiH(4e, 6o)
Basis set: STO-3G
# of qubits: 12
depth D = 8
Molecule: LiH(4e, 6o)
Basis set: STO-3G
# of qubits: 12
Molecule: LiH(4e, 6o)
Basis set: STO-3G
# of qubits: 12
depth D = 20
For results on azobenzene, see our paper on arXiv!
arXiv:2002.11724
automatically generates multiple states finally, diagonalize on a classical computer](https://image.slidesharecdn.com/aps2020postergcp-200303053025/85/Calculating-transition-amplitudes-by-variational-quantum-eigensolvers-1-320.jpg)
Calculating transition amplitudes by variational quantum eigensolvers
- 1. 𝑅 [Å] 𝑅 [Å]𝑅 [Å] Calculating transition amplitudes by variational quantum eigensolvers C71.00241 : Implementation of excited state energy and its analytical derivatives for photochemical reaction simulations on NISQ devices Yohei Ibe1, Yuya O. Nakagawa1,Takahiro Yamamoto1, Kosuke Mitarai2, 1, Tennin Yan1, Qi Gao3, Takao Kobayashi3 1QunaSys Inc., 2Osaka University, 3Mitsubishi Chemical Corp. 1. Methods for excited states based on VQE ü Simulating excited states of molecules is of great concern in photophysics & photochemistry, and there are several methods based on a quantum algorithm, Variational Quantum Eigensolver (VQE) ü Among them, VQD method is remarkably accurate; however, there are no methods to calculate transition amplitudes (i.e., off-diagonal matrix elements) between eigenstates obtained by the VQD ü We propose such a method feasible on NISQ devices and demonstrate on a sampling simulator Summary 3. Calculating transition amplitudes ü VQD is an accurate way to simulate excited states on a quantum computer ü We proposed a method to calculate transition amplitudes between two orthogonal states in a hardware-friendly manner, which is applicable (not only) for the VQD ü This work enlarges the possibility of the VQD and advances the field of excited states calculations on a quantum device 4. Sampling simulation with shot noise 5. Conclusion Appendix: Comparison of ansatz LiH 𝑅 Li H STO-3G (4e, 6o) (12 qubits) Method for excited states Accuracy Easy to calculate transition amplitudes? SSVQE Fair Yes MCVQE Intermediate Yes VQD Excellent No Yes (This work) Diazene (N2H2) 2. Comparison of methods by noiseless simulation Potential energy curve Error from exact calc. O. Higgott et al., Quantum 3, 159 (2019)R. Parrish et al., Phys. Rev. Lett. 122, 230401 (2019) K. Nakanishi et al., Phys. Rev. Research 1, 033062 (2019) ℒ(𝜽) = * +,- . 𝑤+⟨𝜑+ 𝑈3 𝜽 𝐻𝑈 𝜽 𝜑+⟩ (𝑤-> 𝑤7 > ⋯ > 0) ℒ(𝜽) = * +,- . ⟨𝜑+ 𝑈3 𝜽 𝐻𝑈 𝜽 𝜑+⟩ 1. Obtain ground state E; with VQE 2. Execute VQE routine to minimize ℒ- 𝜃 = 𝜓 𝜃 𝐻 𝜓 𝜃 + 𝑤- ⟨E;|𝜓 𝜃 ⟩ 7 → Obtain first excited state E- = 𝜓 𝜃- ∗ 3. Repeat for higher excited states SSVQE Subspace-Search VQE MCVQE Multistate-Contracted VQE VQD Variational Quantum Deflation : physical quantity (Hermitian op.), where . Following equality holds for transition amplitude : Each term can be measured on real devices Calculation setups • Molecular structures: several points along minimum energy path between S2 Franck-Condon (cis/trans) & S2 minimum • Calculation level: 6-31G*/CASCI(3o, 4e), 8 qubits • Ansatz: RSP ansatz (D=20, see Appendix) • High-speed simulator Qulacs [http://qulacs.org/] is used Enables calculation of transition amplitudes with VQD Unable to measure as it is (A is not unitary) Definition of oscillator strength 𝑓+B : 𝛼-coordinate of the 𝑙-th electron where : electric dipole moment operator in atomic units LiH 𝑅 Li H STO-3G CASCI(2e, 2o) (2 qubits) Potential energy curve & error Oscillator strength RY ansatz ü VQD can generate excited states the most accurately ü SSVQE and MCVQE can readily calculate transition amplitudes ü However, VQD has no known methods to calculate transition amplitudes on real NISQ devices ü Transition amplitudes are required for calculating various physical quantities (e.g., oscillator strengths) (𝐴: Hermitian op.) Calculation setups • Calculation level: 6-31G*/CASCI(2o, 2e), 2 qubits (parity mapping is used to reduce the number of qubits) • Ansatz: RY ansatz (D=2) • Two options for the optimization routine for this experiment Blue dots: use sampling simulator in the whole process (including optimization routine) Orange dots: use sampling simulator only for calculating oscillator strengths (parameters are optimized with the noiseless simulator), still using our proposed method Using RSP ansatz (D=10, see Appendix) J. T. Seeley et al., J. Chem. Phys. 137, 224109 (2012) (Available in Qiskit Aqua v0.6.4) Unitary operators Assuming Hardware-efficient ansatz 𝑈FGGHI 𝜽 = exp 𝑇HI(𝜽) − 𝑇HI 3 (𝜽) 𝑇HI 𝜽 = * +:PQQRSTUV W:XTYZR[ 𝜃+ W 𝑐W 3 𝑐+ + * +B:PQQRSTUV W^:XTYZR[ 𝜃+B W^ 𝑐W 3 𝑐^ 3 𝑐+ 𝑐B A. Kandala et al., Nature 549, 242 (2017) A. Peruzzo et al., Nat. Comm., 5,4213 (2014) ×𝐷 UCC-SD ansatz RSP ansatz (real-valued symmetry preserving ansatz) 𝑅 [Å] 𝑅 [Å] 𝑅 [Å] 𝑅 [Å] Energy[Ha] For original symmetry-preserving ansatz, see P. Barkoutsos et al., Phys. Rev. A 98, 022322 (2018) Energy[Ha] Energy[Ha] Molecule: LiH(4e, 6o) Basis set: STO-3G # of qubits: 12 depth D = 8 Molecule: LiH(4e, 6o) Basis set: STO-3G # of qubits: 12 Molecule: LiH(4e, 6o) Basis set: STO-3G # of qubits: 12 depth D = 20 For results on azobenzene, see our paper on arXiv! arXiv:2002.11724 automatically generates multiple states finally, diagonalize on a classical computer